Theorem an2anr

Description: Double commutation in conjunction. (Contributed by Peter Mazsa, 27-Jun-2019)

	Ref	Expression
Assertion	an2anr	$$⊢\left(\left(\phi \wedge \psi \right)\wedge \left(\chi \wedge \theta \right)\right)\leftrightarrow \left(\left(\psi \wedge \phi \right)\wedge \left(\theta \wedge \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ancom	$$⊢\left(\phi \wedge \psi \right)\leftrightarrow \left(\psi \wedge \phi \right)$$
2		ancom	$$⊢\left(\chi \wedge \theta \right)\leftrightarrow \left(\theta \wedge \chi \right)$$
3	1 2	anbi12i	$$⊢\left(\left(\phi \wedge \psi \right)\wedge \left(\chi \wedge \theta \right)\right)\leftrightarrow \left(\left(\psi \wedge \phi \right)\wedge \left(\theta \wedge \chi \right)\right)$$